Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,30,23)(20,31,34,39)(21,25,37,44)(24,32,41,28)(29,42,35,38)(33,47,36,48)(40,46,43,45);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,35,37,29)(24,39,41,31)(25,42,44,38)(33,45,36,46)(40,47,43,48);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20,19,28,22,34,30,32)(18,24,23,31,26,41,27,39)(21,36,29,46,37,33,35,45)(25,43,38,48,44,40,42,47);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,22,37)(18,25,26,44)(19,29,30,35)(20,33,34,36)(23,38,27,42)(24,40,41,43)(28,45,32,46)(31,47,39,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(20,34)(21,37)(23,27)(28,32)(29,35)(40,43)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6939064);